Let \(\lambda \in Z , \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{…

Let \(\lambda \in Z , \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{c}\) be a vector such that \((\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=-17\) and \(\vec{b} \cdot \vec{c}=-20\). Then \(|\vec{c} \times(\lambda \hat{i}+\hat{j}+\hat{k})|^2\) is equal to

[JEE Main 2023, 12 Apr (Shift 1)]